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Finite-extensible implies class-preserving

This article gives the statement and possibly, proof, of an implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., finite-extensible automorphism) must also satisfy the second automorphism property (i.e., class-preserving automorphism)
View all automorphism property implications | View all automorphism property non-implications
Get more facts about finite-extensible automorphism|Get more facts about class-preserving automorphism
This fact is related to: Extensible automorphisms problem
View other facts related to Extensible automorphisms problemView terms related to Extensible automorphisms problem |

Contents

Statement

Related facts

Other results towards the associated conjecture/problem

Further information: Extensible automorphisms problem, extensible automorphisms conjecture, finite-extensible implies inner

Other facts about finite groups proved using the same method

Facts about infinite groups proved using similar constructions


Facts used

Proof

Facts (1) and (2) combine to yield that any finite-extensible automorphism is linearly pushforwardable over a (finite) prime field where the prime does not divide the order of the group, and fact (3) yields that if the field chosen is a class-separating field for the group, then the automorphism is class-preserving. Thus, we need to show that for every finite group, there exists a prime field with the prime not dividing the order of the group, such that the field is a class-separating field for the group. This is achieved by facts (4) and (5).